Holomorphic Spaces and Operator Theory

摘要：本课题的核心是系统地研究全纯函数空间与算子理论之间的某些相互作用。这些全纯空间包括Dirichlet空间、包含Bloch空间和BMOA的潜在容量相关尺度空间以及解析Morrey空间。这些算子是乘子，双线性形式，如Hankel算子和Toeplitz算子，以及换向子。研究对象包括：正交投影、点求值算子、均值算子、d-bar等微分算子、拉普拉斯算子和狄拉克算子、Clifford分析、原子分解、近似数、前双元、中间空间、Carleson测度和势容量。本研究将运用及发展谐波分析、复分析及算符理论等技术。全纯函数空间与算子理论的相互作用不仅在纯数学研究中具有独特的魅力，而且在应用科学的许多领域都有广泛的应用。工程中的自动控制、导航和系统设计就是应用实例。算符理论起源于量子力学的严格化，现在被广泛应用于控制理论。

PI: Zhijian, WuProject Number: 0200587ABSTRACT:The core of this proposal is to systematically study certain interactions between holomorphic function spaces and operator theory. These holomorphic spaces include Dirichlet spaces, potential capacity related scale space which covers Bloch space and BMOA, and the analytic Morrey spaces. The operators are multipliers, bilinear forms, such as Hankel operators and Toeplitz operators, and commutators. The study involves the following objects: the orthogonal projection, the point evaluation operator, mean value operator, the differential operator such as d-bar, Laplacian and Dirac operators, Clifford analysis, atomic decomposition, approximation numbers, predual, intermediate spaces, Carleson measures and potential capacity. Techniques in harmonic analysis, complex analysis and operator theory will bebe used and developed in this study. Beside its own charm in pure mathematical research, the interaction between holomorphic function spaces and operator theory has many applications in several fields of applied sciences. Automatic control, navigation and system design in engineering are examples of the applications. Operator theory has its roots in the rigorization of quantum mechanics and it is now widely used in control theory.